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<h1 id="Interface-fitted-Mesh-Generator:-Two-Dimensions">Interface-fitted Mesh Generator: Two Dimensions<a class="anchor-link" href="#Interface-fitted-Mesh-Generator:-Two-Dimensions">&#182;</a></h1><p>We explain a simple mesh generator for generating an interface-fitted mesh in two dimensions.</p>
<p><strong>Algorithm:  2D Interface-fitted Mesh Generation</strong></p>
<ol>
<li><p>Generate a uniform Cartesian mesh of the box</p>
</li>
<li><p>Find points near the interface.</p>
<ul>
<li>cut points</li>
<li>mesh points near or on the interface</li>
<li>centers of special elements.</li>
</ul>
</li>
<li><p>Construct a Delaunay triangulation on these points.</p>
</li>
<li><p>Post processing.</p>
<ul>
<li>remove triangles away from the interface</li>
<li>merge all uncut elements</li>
</ul>
</li>
</ol>
<p><strong>Reference</strong>: L. Chen, H. Wei, M. Wen. <a href="https://www.math.uci.edu/~chenlong/interface_vem.html">An Interface-Fitted Mesh Generator and Virtual Element Methods for Elliptic Interface Problems</a>. <em>Journal of Computational Physics</em>. 334(1), 2017, 327–348.</p>

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<h2 id="Generate-a-uniform-Cartesian-mesh-of-the-box">Generate a uniform Cartesian mesh of the box<a class="anchor-link" href="#Generate-a-uniform-Cartesian-mesh-of-the-box">&#182;</a></h2>
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<div class="prompt input_prompt">In&nbsp;[18]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">box</span> <span class="p">=</span> <span class="p">[</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">];</span>
<span class="n">h</span> <span class="p">=</span> <span class="mf">0.1</span><span class="p">;</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="n">T</span><span class="p">]</span> <span class="p">=</span> <span class="n">squarequadmesh</span><span class="p">(</span><span class="n">box</span><span class="p">,</span><span class="n">h</span><span class="p">);</span>
<span class="n">edge</span> <span class="p">=</span> <span class="n">T</span><span class="p">.</span><span class="n">edge</span><span class="p">;</span>
<span class="n">edge2elem</span> <span class="p">=</span> <span class="n">T</span><span class="p">.</span><span class="n">edge2elem</span><span class="p">;</span>
<span class="n">clear</span> <span class="n">T</span><span class="p">;</span>
<span class="n">N</span> <span class="p">=</span> <span class="nb">size</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
<span class="n">NT</span> <span class="p">=</span> <span class="nb">size</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
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<h2 id="Find-points-near-the-interface">Find points near the interface<a class="anchor-link" href="#Find-points-near-the-interface">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">phi</span> <span class="p">=</span> <span class="p">@</span><span class="n">phiheart</span><span class="p">;</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">hold</span> <span class="n">on</span><span class="p">;</span>

<span class="c">% compute the phi value at each vertex</span>
<span class="n">phiValue</span> <span class="p">=</span> <span class="n">phi</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">phiValue</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">phiValue</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">eps</span><span class="o">*</span><span class="n">h</span><span class="p">)</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>  <span class="c">% treat nearby nodes as on the interface</span>
<span class="n">vSign</span> <span class="p">=</span> <span class="nb">sign</span><span class="p">(</span><span class="n">phiValue</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">vSign</span><span class="o">==</span><span class="mi">0</span><span class="p">);</span>

<span class="c">% Find the intersection points between edges and the interface</span>
<span class="n">isCutEdge</span> <span class="p">=</span> <span class="p">(</span><span class="n">vSign</span><span class="p">(</span><span class="n">edge</span><span class="p">(:,</span><span class="mi">1</span><span class="p">))</span><span class="o">.*</span> <span class="n">vSign</span><span class="p">(</span><span class="n">edge</span><span class="p">(:,</span><span class="mi">2</span><span class="p">))</span><span class="o">&lt;</span><span class="mi">0</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">edge</span><span class="p">,</span><span class="n">isCutEdge</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;draw&#39;</span><span class="p">);</span>
<span class="n">A</span> <span class="p">=</span> <span class="n">node</span><span class="p">(</span><span class="n">edge</span><span class="p">(</span><span class="n">isCutEdge</span><span class="p">,</span><span class="mi">1</span><span class="p">),:);</span>
<span class="n">B</span> <span class="p">=</span> <span class="n">node</span><span class="p">(</span><span class="n">edge</span><span class="p">(</span><span class="n">isCutEdge</span><span class="p">,</span><span class="mi">2</span><span class="p">),:);</span>
<span class="n">cutNode</span> <span class="p">=</span> <span class="n">findintersectbisect</span><span class="p">(</span><span class="n">phi</span><span class="p">,</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">);</span>
<span class="n">Ncut</span> <span class="p">=</span> <span class="nb">size</span><span class="p">(</span><span class="n">cutNode</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span>
<span class="n">vSign</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="o">+</span><span class="n">Ncut</span><span class="p">)</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">cutNode</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;color&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;MarkerSize&#39;</span><span class="p">,</span><span class="mi">12</span><span class="p">);</span>
</pre></div>

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<div class="prompt input_prompt">In&nbsp;[20]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">hold</span> <span class="n">on</span><span class="p">;</span>
<span class="c">% find interface elem and nodes</span>
<span class="n">isInterfaceElem</span> <span class="p">=</span> <span class="n">false</span><span class="p">(</span><span class="n">NT</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>  
<span class="n">isInterfaceElem</span><span class="p">(</span><span class="n">edge2elem</span><span class="p">(</span><span class="n">isCutEdge</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]))</span> <span class="p">=</span> <span class="n">true</span><span class="p">;</span>
<span class="n">isInterfaceElem</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">vSign</span><span class="p">(</span><span class="n">elem</span><span class="p">)),</span> <span class="mi">2</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">)</span> <span class="p">=</span> <span class="n">true</span><span class="p">;</span> <span class="c">% 2 vertices on interface</span>
<span class="n">findelem</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="n">isInterfaceElem</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">);</span>        

<span class="n">isInterfaceNode</span> <span class="p">=</span> <span class="n">false</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
<span class="n">isInterfaceNode</span><span class="p">(</span><span class="n">elem</span><span class="p">(</span><span class="n">isInterfaceElem</span><span class="p">,:))</span> <span class="p">=</span> <span class="n">true</span><span class="p">;</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">isInterfaceNode</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;MarkerSize&#39;</span><span class="p">,</span><span class="mi">12</span><span class="p">);</span>
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<div class="prompt input_prompt">In&nbsp;[21]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">hold</span> <span class="n">on</span><span class="p">;</span>
<span class="c">% add centers of special elements (two vertices on the interface)</span>
<span class="c">%  0 - -1    -1 - 0     0 -  0      1 - 0</span>
<span class="c">%  1 -  0     0 - 1     1 - -1      1 - 0</span>
<span class="n">isSpecialElem</span> <span class="p">=</span> <span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">vSign</span><span class="p">(</span><span class="n">elem</span><span class="p">)),</span><span class="mi">2</span><span class="p">)</span> <span class="o">&lt;</span><span class="p">=</span> <span class="mi">2</span><span class="p">);</span> <span class="c">% two or more vertices on interfaces</span>
<span class="c">% isSpecialElem = (sum(vSign(elem),2) == 0) &amp; (sum(abs(vSign(elem)),2) == 2);</span>
<span class="n">specialElem</span> <span class="p">=</span> <span class="n">elem</span><span class="p">(</span><span class="n">isSpecialElem</span><span class="p">,:);</span>
<span class="n">auxPoint</span> <span class="p">=</span> <span class="p">(</span><span class="n">node</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span><span class="mi">1</span><span class="p">),:)</span> <span class="o">+</span> <span class="n">node</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span> <span class="mi">3</span><span class="p">),:))</span><span class="o">/</span><span class="mf">2.0</span><span class="p">;</span>
<span class="n">Naux</span> <span class="p">=</span> <span class="nb">size</span><span class="p">(</span><span class="n">auxPoint</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
<span class="n">vSign</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="n">Ncut</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="o">+</span><span class="n">Ncut</span><span class="o">+</span><span class="n">Naux</span><span class="p">)</span> <span class="p">=</span> <span class="nb">sign</span><span class="p">(</span><span class="n">phi</span><span class="p">(</span><span class="n">auxPoint</span><span class="p">));</span>
<span class="c">% find the first two cases</span>
<span class="n">isDiagInterface</span> <span class="p">=</span> <span class="p">(</span><span class="n">vSign</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span><span class="mi">1</span><span class="p">))</span><span class="o">.*</span><span class="n">vSign</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span><span class="mi">3</span><span class="p">))</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">|</span> <span class="c">...</span>
                  <span class="p">(</span><span class="n">vSign</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span><span class="mi">2</span><span class="p">))</span><span class="o">.*</span><span class="n">vSign</span><span class="p">(</span><span class="n">specialElem</span><span class="p">(:,</span><span class="mi">4</span><span class="p">))</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">);</span>
<span class="n">vSign</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="n">Ncut</span><span class="o">+</span><span class="nb">find</span><span class="p">(</span><span class="n">isDiagInterface</span><span class="p">))</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>            
<span class="n">findnode</span><span class="p">(</span><span class="n">auxPoint</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;color&#39;</span><span class="p">,</span><span class="s">&#39;m&#39;</span><span class="p">);</span>
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<div class="prompt input_prompt">In&nbsp;[22]:</div>
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    <div class="input_area">
<div class=" highlight hl-matlab"><pre><span></span><span class="c">% add cut points and aux points</span>
<span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="n">node</span><span class="p">;</span> <span class="n">cutNode</span><span class="p">;</span> <span class="n">auxPoint</span><span class="p">];</span>
<span class="n">nearInterfaceNode</span> <span class="p">=</span> <span class="p">[</span><span class="n">node</span><span class="p">(</span><span class="n">isInterfaceNode</span><span class="p">,:);</span> <span class="n">cutNode</span><span class="p">;</span> <span class="n">auxPoint</span><span class="p">];</span>
<span class="n">interfaceNodeIdx</span> <span class="p">=</span> <span class="p">[</span><span class="nb">find</span><span class="p">(</span><span class="n">isInterfaceNode</span><span class="p">);</span> <span class="n">N</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="n">Ncut</span><span class="p">)</span><span class="o">&#39;</span><span class="p">;</span> <span class="n">N</span><span class="o">+</span><span class="n">Ncut</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="n">Naux</span><span class="p">)</span><span class="o">&#39;</span><span class="p">];</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">nearInterfaceNode</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;color&#39;</span><span class="p">,</span><span class="s">&#39;r&#39;</span><span class="p">,</span><span class="s">&#39;MarkerSize&#39;</span><span class="p">,</span><span class="mi">11</span><span class="p">);</span>
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<h2 id="Construct-a-Delaunay-triangulation-on-these-points">Construct a Delaunay triangulation on these points<a class="anchor-link" href="#Construct-a-Delaunay-triangulation-on-these-points">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">% construct the Delaunay triangulation of interfaceNode</span>
<span class="c">% different versions of matlab using different delaunay triangulation</span>
<span class="n">matlabversion</span> <span class="p">=</span> <span class="n">version</span><span class="p">();</span>
<span class="k">if</span> <span class="n">str2double</span><span class="p">(</span><span class="n">matlabversion</span><span class="p">(</span><span class="k">end</span><span class="o">-</span><span class="mi">5</span><span class="p">:</span><span class="k">end</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span> <span class="o">&lt;</span><span class="p">=</span> <span class="mi">2013</span>
    <span class="n">DT</span> <span class="p">=</span> <span class="n">DelaunayTri</span><span class="p">(</span><span class="n">nearInterfaceNode</span><span class="p">);</span> <span class="c">%#ok&lt;*DDELTRI&gt;</span>
    <span class="n">tElem</span> <span class="p">=</span> <span class="n">DT</span><span class="p">.</span><span class="n">Triangulation</span><span class="p">;</span>
<span class="k">else</span>
    <span class="n">DT</span> <span class="p">=</span> <span class="n">delaunayTriangulation</span><span class="p">(</span><span class="n">nearInterfaceNode</span><span class="p">);</span>
    <span class="n">tElem</span> <span class="p">=</span> <span class="n">DT</span><span class="p">.</span><span class="n">ConnectivityList</span><span class="p">;</span>
<span class="k">end</span>
<span class="n">tElem</span> <span class="p">=</span> <span class="n">fixorder</span><span class="p">(</span><span class="n">nearInterfaceNode</span><span class="p">,</span><span class="n">tElem</span><span class="p">);</span> <span class="c">% correct the orientation</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">tElem</span><span class="p">);</span>
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<p>The Delaunay triangulation is a mesh for the coonvex hull of the given points. Without postprocess, it is messy.</p>
<h2 id="Post-processing">Post-processing<a class="anchor-link" href="#Post-processing">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">% get rid of the unnecessary triangles</span>
<span class="n">NI</span> <span class="p">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">isInterfaceNode</span><span class="p">);</span>  <span class="c">% number of pts of the background mesh near interface</span>
<span class="n">haveNewPts</span> <span class="p">=</span> <span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">tElem</span> <span class="o">&gt;</span> <span class="n">NI</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">);</span> <span class="c">% triangles containing new added vertices</span>
<span class="n">tElem</span> <span class="p">=</span> <span class="n">tElem</span><span class="p">(</span><span class="n">haveNewPts</span><span class="p">,:);</span> 
<span class="n">tElem</span> <span class="p">=</span> <span class="n">interfaceNodeIdx</span><span class="p">(</span><span class="n">tElem</span><span class="p">);</span>  <span class="c">% map interfaceNode index to node index</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">tElem</span><span class="p">,</span><span class="s">&#39;Facecolor&#39;</span><span class="p">,</span><span class="s">&#39;y&#39;</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">vSign</span><span class="o">==</span><span class="mi">0</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">);</span>
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<div class="prompt input_prompt">In&nbsp;[25]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">% get the remainding quad elems</span>
<span class="n">sElem</span> <span class="p">=</span> <span class="n">elem</span><span class="p">(</span><span class="o">~</span><span class="n">isInterfaceElem</span><span class="p">,:);</span>
<span class="c">% merge into one triangulation</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="n">tElem</span><span class="p">;</span> <span class="n">sElem</span><span class="p">(:,[</span><span class="mi">2</span> <span class="mi">3</span> <span class="mi">1</span><span class="p">]);</span> <span class="n">sElem</span><span class="p">(:,[</span><span class="mi">4</span> <span class="mi">1</span> <span class="mi">3</span><span class="p">])];</span>
<span class="n">T</span> <span class="p">=</span> <span class="n">auxstructure</span><span class="p">(</span><span class="n">tElem</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">hold</span> <span class="n">on</span><span class="p">;</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">tElem</span><span class="p">,</span><span class="s">&#39;Facecolor&#39;</span><span class="p">,</span><span class="s">&#39;y&#39;</span><span class="p">);</span>
<span class="n">isInterfaceEdge</span> <span class="p">=</span> <span class="p">((</span><span class="n">vSign</span><span class="p">(</span><span class="n">T</span><span class="p">.</span><span class="n">edge</span><span class="p">(:,</span><span class="mi">1</span><span class="p">))</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">vSign</span><span class="p">(</span><span class="n">T</span><span class="p">.</span><span class="n">edge</span><span class="p">(:,</span><span class="mi">2</span><span class="p">))</span> <span class="o">==</span> <span class="mi">0</span><span class="p">);</span>
<span class="n">interfaceEdge</span> <span class="p">=</span> <span class="n">T</span><span class="p">.</span><span class="n">edge</span><span class="p">(</span><span class="n">isInterfaceEdge</span><span class="p">,:);</span>
<span class="n">nearInterfaceNode</span> <span class="p">=</span> <span class="nb">find</span><span class="p">(</span><span class="n">vSign</span> <span class="o">==</span> <span class="mi">0</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">interfaceEdge</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;draw&#39;</span><span class="p">);</span>
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>
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<div class="prompt input_prompt">In&nbsp;[26]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Generate interface data</span>
<span class="n">interface</span><span class="p">.</span><span class="n">vSign</span> <span class="p">=</span> <span class="n">vSign</span><span class="p">;</span>
<span class="n">interface</span><span class="p">.</span><span class="n">tElem</span> <span class="p">=</span> <span class="n">tElem</span><span class="p">;</span>
<span class="n">interface</span><span class="p">.</span><span class="n">sElem</span> <span class="p">=</span> <span class="n">sElem</span><span class="p">;</span>
<span class="n">interface</span><span class="p">.</span><span class="n">edge</span> <span class="p">=</span> <span class="n">interfaceEdge</span><span class="p">;</span>
<span class="n">interface</span><span class="p">.</span><span class="n">node</span> <span class="p">=</span> <span class="n">nearInterfaceNode</span><span class="p">;</span>
</pre></div>

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<h2 id="Further-Reading">Further Reading<a class="anchor-link" href="#Further-Reading">&#182;</a></h2><p>Run the following example in MATLAB and read <code>interfacemeshdoc</code> to learn more.</p>

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<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
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    <div class="input_area">
<div class=" highlight hl-matlab"><pre><span></span><span class="n">box</span> <span class="p">=</span> <span class="p">[</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">];</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="n">interface</span><span class="p">]</span> <span class="p">=</span> <span class="n">interfacemeshdoc</span><span class="p">(</span><span class="n">box</span><span class="p">,@</span><span class="n">phiheart</span><span class="p">,</span><span class="mf">0.05</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">interface</span><span class="p">.</span><span class="n">edge</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">,</span><span class="s">&#39;draw&#39;</span><span class="p">);</span>
</pre></div>

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